Continuum Mechanics

continuum mechanics

Continuum Mechanics

the branch of mechanics devoted to the study of the motion and equilibrium of gases, liquids, and deformable solids. Subdivisions of continuum mechanics include hydroaeromechanics, gas dynamics, elasticity theory, and plasticity theory. The main assumption of continuum mechanics is that matter can be considered as a continuous medium with its molecular (atomic) structure disregarded and that the distribution of all the characteristics of the medium (including density, stresses, and velocities of particles) can also be considered to be continuous. This is justified by the negligibility of the dimensions of molecules in comparison with the dimensions of the particles that are considered in theoretical and experimental studies in continuum mechanics. Therefore, the apparatus of higher mathematics, which has been well developed for continuous functions, may be employed in continuum mechanics.

The following are the basic equations in the study of any medium in continuum mechanics: (1) the equations of motion or equilibrium for the medium, which are obtained as a consequence of the fundamental laws of mechanics; (2) the continuity equation for the medium, which is a consequence of the law of conservation of mass; and (3) the energy equation. The distinctive features of each specific medium are taken into account by the equation of state or by the rheological equation; the latter establishes for a given medium the form of the relation among the stresses or their rates of change and the strains or their rates of change for the particles. The characteristics of the medium may also depend on the temperature and other physicochemical parameters; the form of these dependences must be established independently. Moreover, the initial and boundary conditions, whose form also depends on the features of the medium, must be specified in solving each particular problem.

Continuum mechanics has an enormous number of important applications in various fields of physics and engineering.

In particular, it can only be achieved by combining my expertise in multiscale computer simulations of solvated polymers with the statistical and continuum mechanics of soft matter structures and dynamics.

One can conclude that the finite element method--being an orthodox daughter of continuum mechanics, where the notion of a point force is forbidden since it leads to singularity--would give infinite displacements for infinitely fine mesh.

The 14 papers include discussions of a two-dimensional variant of a theorem of Uraltseva and Urdaletova for higher-order variational problems, smooth approximations of solutions to nonconvex fully nonlinear elliptic equations, current and curvature varifolds in continuum mechanics, the fundamental solution of an elliptic equation in nondivergence form, the attainability of infima in the critical Sobolev trace embedding theorem on manifolds, and the global solvability of Navier-Stokes equations for a non-homogeneous non-Newtonian fluid.

This paradigm shift, from continuum mechanics to (lattice) kinetic theory, brings about a series of major computational assets, primarily the fact that non-locality (particle streaming) is linear and non-linearity (particle collisions) is local.

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